group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
The twisted cohomology-version of de Rham cohomology (a simple example of twisted differential cohomology):
For $X$ a smooth manifold and $H \in \Omega^3(X)$ a closed differential 3-form, the $H$ twisted de Rham complex is the $\mathbb{Z}_2$-graded vector space $\Omega^{even}(X) \oplus \Omega^{odd}(X)$ equipped with the $H$-twisted de Rham differential
Notice that this is nilpotent, due to the odd degree of $H$, such that $H \wedge H = 0$, and the closure of $H$, $d H = 0$.
There is also the cohomology of the chain complex whose maps are just multiplication by $H$.
This is also called H-cohomology (Cavalcanti 03, p. 19).
If the de Rham complex $(\Omega^\bullet(X),d_{dr})$ is formal, then for $H \in \Omega_{cl}^3(X)$ a closed differential 3-form, the $H$-twisted de Rham cohomology of $X$ coincides with its H-cohomology, for any closed 3-form $H$.
The concept of $H_3$-twisted de Rham cohomology was introduced (in discussion of the B-field in string theory), in:
Further discussion:
Peter Bouwknegt, Alan Carey, Varghese Mathai, Michael Murray, Danny Stevenson, Section 9.3 of: Twisted K-theory and K-theory of bundle gerbes , Commun Math Phys, 228 (2002) 17-49 (arXiv:hep-th/0106194, doi:10.1007/s002200200646)
Varghese Mathai, Danny Stevenson, Section 3 of: Chern character in twisted K-theory: equivariant and holomorphic cases, Commun. Math. Phys. 236:161-186, 2003 (arXiv:hep-th/0201010)
Daniel Freed, Michael Hopkins, Constantin Teleman, Section 2 of: Twisted equivariant K-theory with complex coefficients, Journal of Topology, Volume 1, Issue 1, 2007 (arXiv:math/0206257, doi:10.1112/jtopol/jtm001)
Constantin Teleman, around Prop. 3.7 in: K-theory of the moduli of bundles over a Riemann surface and deformations of the Verlinde algebra, in: Ulrike Tillmann (ed.) Topology, geometry and quantum field theory, Cambridge 2004 (arXiv:math/0306347, spire:660158)
Gil Cavalcanti, Section I.4 of: New aspects of the $d d^c$-lemma, Oxford 2005 (arXiv:math/0501406)
The classical case of twisted de Rham cohomology with twists in degree 1, given by flat connections on flat line bundles and more generally on flat vector bundles), and its equivalence to sheaf cohomology with coefficients in abelian sheaves of flat sections (local systems):
Review:
Cailan Li, Cohomology of Local Systems on $X_\Gamma$ (pdf, pdf)
Youming Chen, Song Yang, Section 2.1 in: On the blow-up formula of twisted de Rham cohomology. Annals of Global Analysis and Geometry volume 56, pages 277–290 (2019) (arXiv:1810.09653, doi:10.1007/s10455-019-09667-8)
Hisham Sati, A Higher Twist in String Theory, J. Geom. Phys. 59:369-373, 2009 (arXiv:hep-th/0701232)
Varghese Mathai, Siye Wu, Analytic torsion for twisted de Rham complexes, J. Diff. Geom. 88:297-332, 2011 (arXiv:0810.4204)
(relation to analytic torsion)
In the broader context of twisted cohomology theory and the Chern-Dold character map:
Last revised on July 25, 2021 at 09:06:58. See the history of this page for a list of all contributions to it.